On the best coapproximation problem in ${\ell }_1^n$

Abstract

We study the best coapproximation problem in the Banach space ${\ell }_1^n,$ by using Birkhoff-James orthogonality techniques. Given a subspace $\mathbb{Y}$ of ${\ell }_1^n$, we completely identify the elements $x$ in ${\ell }_1^n,$ for which best coapproximations to $x$ out of $\mathbb{Y}$ exist. The methods developed in this article are computationally effective and it allows us to present an algorithmic approach to the concerned problem. We also identify the coproximinal subspaces and co-Chebyshev subspaces of ${\ell }_1^n$.